The long-term return rate to a stable equilibrium (sometimes called ‘asymptotic resilience’ ) is approximated by the dominant eigenvalue of the Jacobian matrix of the model representing the system.

its return rate to a stable equilibrium state following a perturbation decreases. As a bifurcation is approached, the system exhibits critical slowing down, i.e. In the situation where the change in an exogenous forcing variable is slow relative to the characteristic speed of the internal variables, the tipping point may be described mathematically as a bifurcation. Awareness of this phenomenon will enrich understanding of the dynamics of complex systems on the verge of a critical transition.Īnticipating abrupt shifts in system state that are caused by gradual directional change in system conditions (critical transitions or ‘tipping points’) is a key issue for the natural and social sciences, including climate science, ecology, finance and epidemiology. Greater attention should be focused on the potential for systems to exhibit transient amplification of perturbations as a critical threshold is approached, and should be considered when searching for generic leading indicators of tipping points. Furthermore, we introduce indicators designed to detect trends in these measures and find that they reliably classify time series of case notifications simulated from stochastic models according to levels of vaccine uptake.

Here, we show that measures of transient dynamics, specifically, reactivity and the maximum of the amplification envelope, also change systematically as a bifurcation is approached in an important class of models for epidemics of infectious diseases. In continuous systems, critical slowing down is mathematically described by a decrease in magnitude of the dominant eigenvalue of the Jacobian matrix on the approach to a critical transition. The majority of known early warning indicators of critical transitions rely on asymptotic resilience and critical slowing down.

Wolfram mathematica matrix maximum